Semi-global weak stabilization of bilinear Schrödinger equations

Abstract We consider a linear Schrodinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times ( t n ) n ∈ N for which the values of the solution of the closed loop system converge weakly in H 2 to the ground state. Here, we prove the convergence of the whole solution, as t → + ∞ . The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.